Optimized defects in band-gap waveguides

ABSTRACT

Disclosed is a photonic band-gap crystal waveguide having the physical dimension of the photonic crystal lattice and the size of the defect selected to provide for optimum mode power confinement to the defect. The defect has a boundary which has a characteristic numerical value associated with it. The ratio of this numerical value to the pitch of the photonic crystal is selected to avoid surface modes found to exist in certain configurations of the photonic band-gap crystal waveguide. Embodiments in accord with the invention having circular and hexagonal defect cross sections are disclosed and described. A method of making the photonic band-gap crystal waveguide is also disclosed and described.

[0001] This is a continuation of U.S. patent application Ser. No.10/067,644 filed on Feb. 4, 2002, the content of which is relied uponand incorporated herein by reference in its entirety, and the benefit ofpriority under 35 U.S.C. § 120 is hereby claimed as well as the benefitand priority to U.S. Provisional Patent Application No. 60/277,312,filed Mar. 20, 2001.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The present invention relates generally to photonic band-gapcrystal waveguides, and particularly to photonic band-gap crystalwaveguides having a low refractive index core region. The invention isalso directed to a method of making low refractive index core photonicband-gap crystal waveguides.

[0004] 2. Technical Background

[0005] Knowledge of how to guide light in a material by means of totalinternal reflection is old in physical science. One of the drawbacks oflight guides using total internal reflection lies in the very principleof total internal reflection. That is, total internal reflection occursat the interface between a first and a second material having differentrefractive indexes. Light traveling in the material of higher refractiveindex is reflected (totally reflected for incident angles lower than thecritical angle) at the interface with the material of lower refractiveindex. Thus, the total internal reflection mechanism acts to confine thelight to the higher index material. The higher index material typicallyis higher in density and so is characterized by higher attenuation dueto Rayleigh scattering and by a higher non-linear coefficient. Thenon-linear effects can be mitigated by designing total internalreflection waveguides that have relatively high effective area. However,the complexity of the core refractive index profile usually increasesfor designs that provide larger effective area. This complexity usuallytranslates to higher cost.

[0006] More recently, diffraction has been studied as a means to guidelight in a material. In a light guiding protocol in which theconfinement mechanism is diffraction, the material in which the light isguided, i.e., the core of the optical waveguide, can have a relativelylow refractive index and thus a lower density. In fact, the use of a gasor a vacuum as a waveguide core becomes practical.

[0007] A particular structure well suited for use as a diffraction typeoptical waveguide is a photonic band-gap crystal. The photonic crystalitself is a regular lattice of features in which the spacing of thefeatures is of the order of the light wavelength to be guided. Thephotonic crystal can be constructed of a first material having a firstrefractive index. Embedded in this first material, in the form of aregular lattice or array, is a second material having a secondrefractive index. This is the basic photonic crystal structure.Variations on this basic design can include more than two materials inthe make up of the photonic band-gap crystal. The number of usefulvariations in the details of the lattice structure is also large. In thebasic photonic crystal structure, the second material can simply bepores or voids formed in the first material. Depending upon therefractive index difference of the materials and the spatial arrangementand pitch (center to center distance between features) of the embeddedfeatures, the photonic crystal will not propagate light having awavelength within a certain wavelength band. This is the “band-gap” ofthe photonic crystal and is the property of the photonic crystal thatprovides for light confinement. It is due to this property that thestructure is given the name, photonic band-gap crystal.

[0008] To form an optical waveguide (or more generally, a structure thatguides electro-magnetic energy), a defect is formed in the photonicband-gap crystal. The defect is a discontinuity in the lattice structureand can be a change in pitch of the lattice, the replacement of aportion of the lattice by a material of different refractive index, orthe removal of a portion of the photonic band-gap crystal material. Theshape and size of the defect is selected to produce or support a mode oflight propagation having a wavelength that is within the band-gap of thephotonic crystal. The walls of the defect are thus made of a material, aphotonic band-gap crystal, which will not propagate the mode produced bythe defect. In analogy with the total internal reflection opticalwaveguide, the defect acts as the waveguide core and the photonicband-gap crystal acts as the clad. However, the mechanism of thewaveguide allows the core to have a very low refractive index thusrealizing the benefits of low attenuation and small non-linearcoefficient.

[0009] Because of the potential benefits provided by a photonic band-gapcrystal waveguide, there is a need to identify defect structures thatproduce modes that have useful wavelengths, the modes being efficientlypropagated over practical distances. More particularly, there is a needto investigate whether photonic band-gap crystal defect structures existthat will allow photonic band-gap crystal waveguides to propagate lightsignals over distances compatible with telecommunication systems.

[0010] Other uses of the photonic band-gap crystal waveguide includethose that involve the delivery of high electromagnetic power levelssuch as in devices for excising material or welding material.

SUMMARY OF THE INVENTION

[0011] One aspect of the present invention is a photonic band-gapcrystal optical waveguide which includes a photonic crystal having aband-gap. Typically, the photonic band-gap crystal is characterized by apitch, the center to center distance between repeating features thatmake up the photonic crystal lattice. The photonic band-gap crystal hasa defect, that is, a break or discontinuity in the regularity of thelattice. The defect is characterized by a boundary enclosing a planecross section of the defect. The enclosing boundary is the locus ofpoints in a plane where the photonic band-gap crystal structure abutsthe defect. Perpendicular to the plane cross section is a characteristiclength dimension of the defect. In the case disclosed and describedherein of a photonic band-gap crystal waveguide structure, the defectlength dimension extends through the photonic band-gap crystal so thatone has access to either end of the defect.

[0012] The boundary of the defect is characterized by a numerical value,which can have units of length. The numerical value can be, for example,a radius, if the defect cross section is circular, the distance of aboundary point from a feature in the cross section (such as thegeometrical center), or the perimeter measure of the boundary. Thenumerical value characteristic of the defect boundary is such thatlocalized modes produced by (supported in) the defect propagate in thewavelength range in the band-gap of the photonic band-gap crystal.Further, the ratio of the numerical value to the photonic band-gapcrystal pitch is selected so that the excitation of surface modes withinthe photonic band-gap is avoided.

[0013] When the defect boundary, together with the photonic band-gapcrystal pitch are such that surface modes are excited or supported(exist), a large fraction of light power propagated along the defect isessentially not located in the defect. The surface mode propagates atleast partially in the photonic band-gap crystal itself. Thus, thedistribution of light power is not effective to realize the benefitsassociated with the low refractive index core of a photonic band-gapcrystal optical waveguide.

[0014] In an embodiment of this first aspect of the invention, thedefect has a circular cross section and the numerical value is theradius of the circle. The ratio of radius to pitch has a range from 0.75to 1.15.

[0015] In a further embodiment of the first aspect of the invention, theratio of radius to pitch is 1.3 to 1.5. In yet another embodiment inaccord with a circular defect cross section, the ratio of radius topitch is 1.7 to 2.1. At ratios between the ranges given in thesecircular cross section embodiments, surface modes appear, drawing lightpower out of the defect.

[0016] This first aspect of the invention and the embodiments thereofcan advantageously be characterized by a defect which is eitherpartially or entirely a void in the photonic band-gap crystal. As analternative, this first aspect if the invention and the embodimentsthereof can be characterized by a defect which is either partially orentirely a material which has a refractive index lower than at least oneof the materials that form the photonic band-gap crystal lattice. As isknown in the art, the photonic band-gap crystal lattice is generallyformed from at least two materials which differ from one another inrefractive index. In a single mode waveguide embodiment in accord withthe first aspect of the invention, the photonic band-gap crystalincludes air. For example, the crystal lattice can be symmetricallyspaced voids or pores formed in a material such as SiO₂. The materialsused to form photonic band-gap crystals are known in the art and aredescribed for example in Photonic Crvstals: Molding the Flow of Light,J. D. Joannopoulos, et al., Princeton University Press, Princeton, 1995.The fractional volume of air making up the photonic band-gap crystal canbe specified as having a particular value or range of values. The termfractional volume of air is the ratio of the volume of the crystal thatis air to the total volume of the crystal. The fractional volume of thepores that may make up the photonic crystal is also a useful measure. Inthis case, the pores may be filled with air, be evacuated, or filledwith a material having a pre-selected refractive index.

[0017] In an embodiment in accord with the invention, the fractionalvolume of air is not less than 0.67, the defect has a circular crosssection and the numerical value characteristic of the defect boundary isthe cross section radius. To achieve a light mode propagating with notless than 0.5 of the mode power in the defect (the mode power fraction),the ratio of radius to pitch is in the range from about 0.61 to 1.22. Toachieve a mode power fraction in the defect of not less than 0.7, theratio of radius to pitch has a range from about 0.63 to 1.19. To achievea mode power fraction not less than 0.8, the ratio of radius to pitchhas a range from about 0.8 to 1.16.

[0018] A mode power fraction not less than 0.9 can be achieved in aphotonic band-gap crystal having a defect of circular cross section anda fractional volume of air not less than 0.83, with a ratio of radius topitch having a range from 1.07 to 1.08. This particular embodiment ofthe waveguide in accord with the invention is single mode.

[0019] In a further embodiment in accord with this aspect of theinvention, the defect cross section is a void of hexagonal crosssection, the photonic band-gap crystal includes pores having volumefraction not less than 0.67. The numerical value associated with thedefect is the length of a line drawn from the center of the hexagonperpendicular to a side of the hexagon. For a mode power fraction withinthe defect not less than 0.6, the ratio of the numerical value to pitchhas a range from 0.9 to 1.35. For mode power within the defect (modepower confinement fraction) not less than 0.8, the ratio of numericalvalue to pitch has a range from 1.45 to 1.65.

[0020] A second aspect of the invention is a method of making a photonicband-gap crystal optical waveguide. The method in accord with theinvention includes the steps of a) fabricating a photonic band-gapcrystal having a pitch; and, b) forming a defect in the photonicband-gap crystal. The defect has a boundary enclosing the defect crosssection and a length perpendicular to the defect cross section. Thedefect can be located within, or partially within, the photonic band-gapcrystal. The boundary is characterized by a numerical value, which isselected such that the wavelength of the localized mode produced by(supported in) the defect propagates in the wavelength range of thephotonic crystal band-gap. The ratio of the numerical valuecharacteristic of the defect boundary to the photonic band-gap crystalpitch is selected to avoid the excitation of surface modes within thephotonic band-gap.

[0021] In an embodiment in accord with the method, the defect is formedby removing material from the photonic band-gap crystal. That is, thedefect is a void in the photonic band-gap crystal.

[0022] In another embodiment of the method the photonic band-gap crystalis made by forming pores or voids in a material. In a further limitationof this embodiment, the voids or pores make up not less than 0.67, andpreferably not less than 0.83, of the volume of the photonic band-gapcrystal.

[0023] Additional features and advantages of the invention will be setforth in the detailed description which follows, and in part will bereadily apparent to those skilled in the art from that description orrecognized by practicing the invention as described herein, includingthe detailed description which follows, the claims, as well as theappended drawings.

[0024] It is to be understood that both the foregoing generaldescription and the following detailed description are merely exemplaryof the invention, and are intended to provide an overview or frameworkfor understanding the nature and character of the invention as it isclaimed. The accompanying drawings are included to provide a furtherunderstanding of the invention, and are incorporated in and constitute apart of this specification. The drawings illustrate various embodimentsof the invention, and together with the description serve to explain theprinciples and operation of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

[0025]FIG. 1 is an illustration of a photonic band-gap crystal having adefect of circular cross section.

[0026]FIG. 2 is an illustration of a photonic band-gap crystal having adefect of circular cross section.

[0027]FIG. 3 is an illustration of a photonic band-gap crystal having adefect of circular cross section.

[0028]FIG. 4 is a chart of energy fraction of the propagated light modewithin the defect versus numerical value to pitch ratio for a defecthaving a circular cross section.

[0029]FIG. 5 is an illustration of a photonic band-gap crystal having adefect of hexagonal cross section.

[0030]FIG. 6 is an illustration of a photonic band-gap crystal having adefect of hexagonal cross section.

[0031]FIG. 7 is an illustration of a photonic band-gap crystal having adefect of hexagonal cross section.

[0032]FIG. 8 is a chart of energy fraction of the propagated light modewithin the defect versus numerical value to pitch ratio for a defecthaving a hexagonal cross section.

[0033]FIG. 9 is a scanning electron microscope photograph of a photonicband-gap crystal having a circular defect.

[0034]FIG. 10 is a scanning electron microscope photograph of a photonicband-gap crystal having a circular defect.

[0035]FIG. 11 is an illustration of the mode power distribution in aphotonic band-gap crystal having a defect for the case of a mode havinga large energy fraction within the defect and for the case of a surfacemode.

DETAILED DESCRIPTION OF THE INVENTION

[0036] Reference will now be made in detail to the present preferredembodiments of the invention, examples of which are illustrated in theaccompanying drawings. Wherever possible, the same reference numberswill be used throughout the drawings to refer to the same or like parts.An exemplary embodiment of the photonic band-gap crystal waveguide ofthe present invention is shown in FIG. 1, which is an end view of thewaveguide structure. The photonic band-gap crystal 10 is illustrated asa lattice of light colored cylinders 3 embedded in a materialrepresented by the dark areas surrounding each of the cylinders.Although the repeating features, cylinders 3, are shown as having acircular cross section, the cross section can in practice have a generalshape, including polygonal, elliptical, or erose. The pitch of thephotonic band-gap crystal 10 is shown as line 4 drawn between thecenters of nearest neighbor features, in this case cylinders. Thephotonic band-gap crystal defect 12 is illustrated as having a circularcross section of radius 2. As in the case of the photonic band-gapcrystal features, the defect cross section can have a general shape andprovide desired performance in accord with the photonic band-gap crystalwaveguide of the invention. The length of defect 12 extends into thephotonic band-gap crystal in a direction perpendicular to the circularcross section. The boundary between the defect and the photonic band-gapcrystal is a circle in this case and the numerical value associated withthe boundary is radius 2. The ratio of radius 2 to pitch 4 is selectedto be in a range to avoid excitation of surface modes within thephotonic band-gap.

[0037] The discovery that surface modes can exist at the boundarybetween a photonic band-gap crystal and a defect therein is a key to thedesign and fabrication of photonic band-gap crystal waveguides that areefficient and practical in a telecommunications environment as well asin environments that include the delivery of high power electro-magneticwaves.

[0038] To calculate the modes supported and the mode power distributionin the photonic band-gap crystal waveguide, Maxwell's vector waveequation having a position dependent dielectric function must be solved.A useful form of this wave equation is found at page 11, equation (7) ofthe Joannopoulos et al. reference cited above. Techniques for solvingthe governing equations are known in the art and appear for example inthe publications: Steven G. Johnson and J. D. Joannopoulos,“Block-iterative frequency-domain methods for Maxwell's equations in aplanewave basis,” Optics Express 8, no. 3, 173-190 (2001). In thispublication, the authors summarize their work as: “Fully-vectorialeigenmodes of Maxwell's equations with periodic boundary conditions werecomputed by preconditioned conjugate-gradient minimization of the blockRayleigh quotient in a planewave basis, using a freely availablesoftware package.” The freely available software package to which theyrefer is set forth in, Steven G. Johnson and J. D. Joannopoulos, The MITPhotonic-Bands Package, and is available on the internet at UniversalResource Identifier http://ab-initio.mit.edu/mpb/.

[0039] The results of the calculation as applied to a photonic band-gapcrystal waveguide illustrated in FIG. 1 is shown as curve segment 14 inFIG. 4. The vertical axis of FIG. 4 is the fraction of the mode energycontained in the defect of the photonic band-gap crystal waveguide. Thehorizontal axis is the ratio of defect radius (numerical value of theboundary) to pitch. Curve segment 14 shows that mode energy fraction inthe defect is a maximum at a ratio of about 1. The local minimum ofcurve 14 located near a ratio of 1.3 corresponds to a defect geometrythat supports, i.e., propagates a surface mode. In alternative language,the defect geometry allows excitation of surface modes within thephotonic band-gap.

[0040]FIG. 2 is an embodiment of the invention essentially identical tothat of FIG. 1 except that defect 12 is characterized by a ratio ofabout 1.5. The propagation characteristics of the photonic band-gapcrystal waveguide of FIG. 2 is shown in curve segment 16 of FIG. 4. Inthe FIG. 2 embodiment, the fraction of mode energy confined to thedefect represented in curve segment 16 is maximum at a radius to pitchratio near 1.5. The local minimum of curve segment 16 that occurs near1.6 is a photonic band-gap crystal waveguide configuration that supportsor allows excitation of one or more surface modes.

[0041] As the ratio continues to increase, i.e., the defect increases insize and the boundary moves out farther into the photonic band-gapcrystal, the fraction of mode energy confined to the defect continues topass through local minima and maxima. FIG. 3 is an embodiment of theinvention in which the defect radius to pitch ratio is about 2, which isthe location of the maximum of curve segment 18 in FIG. 4. Curve segment18 corresponds to the photonic band-gap crystal waveguide illustrated inFIG. 3.

[0042] For a desired fraction of confined mode energy, the range ofallowed ratios can be read from the appropriate curve segment, 14, 16,or 18 in FIG. 4. For the smallest ratio embodiment of a photonicband-gap crystal waveguide in accord with the invention, illustrated inFIG. 1, the waveguide is single mode and the optimum ratio provides fora fraction of mode energy confined to the defect near 0.8. The fractionof confinement is higher as defect radius 2 increases and additionalmodes are propagated in the defect.

[0043] Additional embodiments of the photonic band-gap crystal waveguidein accord with the invention, and having a defect 20 of hexagonal crosssection are shown in FIGS. 5-7. The photonic band-gap crystal 10 in eachof these embodiments is essentially identical to that of FIGS. 1-3. Theratio of numerical value 22 to pitch is about 1 in FIG. 5, where thenumerical value 22 is defined as the perpendicular distance from thecenter of the hexagonal defect cross section to one of the sides of thehexagon. In the case of FIGS. 6 and 7, the value of the respectiveratios of numerical value 22 to pitch is 1.5 and 2.0. The chart of FIG.8 shows the fraction of mode energy confined to the defect. Curvesegment 24 has a maximum of confinement near 0.8 at a ratio near 1.2.Another maximum having a confinement fraction near 0.9 is shown by curvesegment 26 and is seen to occur at a ratio of about 1.5.

[0044] Similar calculations can be carried out for essentially anyconfiguration of photonic band-gap crystal having a defect ofessentially any cross section.

[0045] The photonic band-gap crystal waveguide can be made using any ofa number of methods known in the art. The methods allow the skilledpractitioner to make a wide range of shapes of the photonic band-gapcrystal features as well as the defect cross section.

[0046] One exemplary method of making a photonic band-gap crystalincludes the step of arranging a plurality of hollow rods into a bundle.The rods are characterized by an inside and an outside radius and thepitch of the photonic band-gap crystal is the distance between centersof nearest neighbor rods in the bundle. The rods can advantageously bemade of a silica base glass. The rods may be held in position by placingthem within a tube, by banding, or by heat treatment. For example thebundle could be held in a fixture while the component rods are heatedsufficiently to cause them to adhere one to another. A glass frit couldalso be used to weld the rods into the desired positions. The pitch ofthe photonic band-gap crystal formed in this manner can be reduced to adesired value by heating and drawing the rod bundle in much the same waya glass preform is heated and drawn to form a waveguide fiber. It isunderstood that the glass rods can be fabricated to have a wide range ofinterior cross sections in combination with a wide range of outsidecross section shapes. An example is a rod having a hexagonal outsidecross section and a circular interior cross section shape. The photonicband-gap crystal waveguides shown in FIGS. 1-3 and 5-7 can be fabricatedusing rods having a circular interior cross section and a circularoutside cross section.

[0047] The defect in the photonic band-gap crystal waveguide can beformed by using a fixture, having a desired cross section, about whichthe rods are assembled. A fixture having essentially any cross sectioncan be used in forming the defect. As an alternative method of formingthe defect, rods can simply be left out of the assembly. As a furtheralternative method of forming the defect, the photonic band-gap crystalcan be etched or machined after bundling is completed.

[0048] An alternative method of forming the photonic band-gap crystal isby extrusion. In this case, by using the proper dies, the photoniccrystal structure and the defect are formed without need for a weldingstep, a holding tube or fixture, or a fixture to form the defect. Theextruded photonic band-gap crystal waveguide can be treated as awaveguide fiber preform and drawn to achieve a desired pitch togetherwith a desired defect numerical value.

[0049] In FIGS. 9 and 10 are shown scanning electron micro-graphs ofphotonic band-gap crystal waveguides made using an assembly of hollowrods arranged about a central tube and enclosed by an outer tube. Therods have respective circular outside and interior cross sections.

[0050] An illustration of mode energy distributions calculated forphotonic band-gap crystal waveguides is given in FIG. 11. Mode energydistribution 28 is a distribution in accord with the invention. The darkcentral portion of distribution 28 represents a concentration of modeenergy at the defect center. Distribution 28 shows a large fraction ofthe mode energy confined to the defect. In contrast, the mode energydistribution 30 is characteristic of a surface mode, i.e., a mode thatexists across the interface between the defect and the body of thephotonic crystal. Mode energy distribution 30 shows that the energy islargely propagated outside the defect, and thus none of benefits withregard to low attenuation and minimization of non-linear effects arerealized.

[0051] It will be apparent to those skilled in the art that variousmodifications and variations of the present invention can be madewithout departing from the spirit and scope of the invention. Thus, itis intended that the present invention include the modifications andvariations of this invention provided they come within the scope of theappended claims and their equivalents.

We claim:
 1. A photonic band-gap crystal optical waveguide comprising: aphotonic band-gap crystal having a pitch and; a defect, including acore, said defect having a boundary that encloses a plane cross sectionand a length dimension perpendicular to the plane cross section, thedefect boundary including a plurality of protrusions.
 2. The photonicband-gap crystal optical waveguide of claim 1, wherein said defect has astructure such that the mode power fraction confined to said core is notless than 0.6.
 3. The photonic band-gap crystal optical waveguide ofclaim 1, wherein said boundary is selected such that the mode powerfraction confined to said core is not less than 0.6.
 4. The photonicband-gap crystal optical waveguide of claim 1, wherein: said boundary isbeing characterized by a numerical value and the numerical value isselected so that the wavelength of the localized mode produced by thedefect propagates in the wavelength range of the photonic band-gap; andthe ratio of the numerical value of said defect to the pitch is selectedto avoid the excitation of surface modes within the photonic band-gap.5. The photonic band-gap crystal optical waveguide of claim 1, wherein:said boundary is being characterized by a perpendicular distance fromdefect center to the nearest point on the boundary, said distance beingsuch that: (i) that the wavelength of the localized mode produced by thedefect propagates in the wavelength range of the photonic band-gap; andthe ratio of the distance to the pitch is selected to avoid theexcitation of surface modes within the photonic band-gap.
 6. Thephotonic band-gap crystal optical waveguide of claim 5, wherein saiddistance is selected so that the mode power fraction confined to thecore is not less than 0.6.
 7. The photonic band-gap crystal opticalwaveguide of claim 1, wherein said band-gap crystal optical waveguide isan optical fiber, said plurality of protrusions being a plurality ofribs situated along the core surface; and said boundary is beingcharacterized a numerical value, said numerical value being the distancefrom core center to the nearest point on one of said ribs.
 8. Thephotonic band-gap crystal optical waveguide of claim 7, wherein, saidcore has refractive index lower than the refractive index of materialimmediately surrounding said core.
 9. The photonic band-gap crystaloptical waveguide of claim 1 wherein, said defect has a circular crosssection plane with said plurality of ribs protruding from the defectboundary, said boundary is being characterized a numerical value and thenumerical value is the radius of the circular cross section measured tothe ribs.
 10. The photonic band-gap crystal optical waveguide accordingto any of the preceding claims, wherein the number of said protrusionsis 6×N, where N is a positive integer.
 11. The photonic band-gap crystaloptical waveguide of claim 1, wherein said waveguide is single modewaveguide, said defect having a circular cross section with theprotruding ribs, said defect boundary is being characterized a thedistance from the center of said cross-section to the nearest point onsaid boundary, and, for a mode power fraction confined to core of notless than 0.6, the ratio of said distance to pitch has a range fromabout 0.6 to 2.5.
 12. The photonic band-gap crystal optical waveguide ofclaim 11, wherein the mode power fraction confined to said core is notless than 0.75.
 12. The photonic band-gap crystal optical waveguide ofclaim 1 wherein, said defect is a core having a hexagonal cross sectionplane, the mode power fraction confined to said core is not less than0.6 and the defect boundary being characterized by a numerical value,wherein the numerical value is the length of a line drawn from thecenter of the hexagonal crossection perpendicular to a side of thehexagon, and, the ratio of the numerical value to pitch has a range from0.6 to 2.5.
 13. The photonic band-gap crystal optical waveguide of claim12, wherein the mode power fraction confined to said core is not lessthan 0.75.
 14. The photonic crystal optical band-gap waveguidecomprising: photonic band-gap crystal having a pitch; and a defect,including a core, said defect having a boundary that encloses a planecross section and a length dimension perpendicular to the plane crosssection, the defect boundary (i) including a plurality of protrusionsand (ii) being characterized by at least one numerical value, whereinsaid numerical value is measured from defect center to the closest pointon said boundary.
 15. The photonic band-gap crystal optical waveguide ofclaim 14, wherein the mode power fraction confined to said core is notless than 0.6.
 16. The photonic band-gap crystal optical waveguide toclaim 14, wherein the number of said protrusions is 6×N, where N is apositive integer.